Problem: Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{-5k^2 + 10k + 400}{-8k^2 - 144k - 640}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {-5(k^2 - 2k - 80)} {-8(k^2 + 18k + 80)} $ $ x = \dfrac{5}{8} \cdot \dfrac{k^2 - 2k - 80}{k^2 + 18k + 80} $ Next factor the numerator and denominator. $ x = \dfrac{5}{8} \cdot \dfrac{(k + 8)(k - 10)}{(k + 8)(k + 10)}$ Assuming $k \neq -8$ , we can cancel the $k + 8$ $ x = \dfrac{5}{8} \cdot \dfrac{k - 10}{k + 10}$ Therefore: $ x = \dfrac{ 5(k - 10)}{ 8(k + 10)}$, $k \neq -8$